Let $f(x;\theta ) = e^{-(x_i-\theta)}exp(-e^{-x_i-\theta)})$
How do I find the Cramer-Rao lower bound?
the log likelihood is
$l(\theta;x)=\Sigma_{i=1}^n{[-(x_i-\theta )-e^{-(x_i-\theta )}]}$
and taking the derivative of the log likelihood gives:
$\frac{\delta}{\delta\theta}l(\theta;x)=n+\Sigma_{i=1}^n[e^{-(x_i-\theta)}]$
Taking the second derivative gives:
$\frac{\delta^2}{\delta\theta^2}l(\theta;x)=\Sigma_{i=1}^n{e^{-(x_i-\theta)}}$
And the expected value of this is:
$$E[\frac{\delta^2}{\delta\theta^2}l(\theta;x)]=n\int\limits_{-\infty}^\infty \exp(-e^{-(x-\theta)}-2(x-\theta)) \ dx$$
Am I on the right track?